Particularly worth visiting is Beopjusa temple (법주사) at the foot of the mountain. A huge, bronze Buddha stands towering over the temple buildings, with their richly painted exteriors and traditional Korean design. It is truly a stunning sight, and you will want to spend some time wandering around.
From the temple, it is possible to continue hiking upwards, following a gentle trail. It's an easy and popular route, with many other hikers at the weekends. After descending again, enjoy the restaurants and market stalls huddled around the entrance to the trail, selling local specialties such as mushrooms and jujubes.
Chungcheongbuk-do, so it might be worth taking a weekend to explore the area if you can. I went there as part of a day bus trip with a Korean tour guide, and was able to spend a few hours at Songnisan and also visit another location before heading back to Seoul. Being centrally located, Chungcheongbuk-do is easy to get to from anywhere in the country.